How to do beveled curves with changing cross sectional shape along the curve?
You can simplify this by manipulating a straight object by hand and use a curve modifier. It is much easier to change this not bended object using your favourite technique like sculpting, lattice, skin or completely by hand in edit mode. 

Let's start with a curve.


I will then create a linear extrude and give some variation in the cross section. In this case I will do it by hand  and provide a good enough resolution along the axis using loop cut.




Applying the curve modifier will deform the object along the curve. If the longitudinal resolution is not good enough, the deformed shape will not look continuous. Some tweaking or smoothing may be required.